VANDERMONDE_INTERP_2D
Data Interpolation with Polynomials using the Vandermonde Matrix
VANDERMONDE_INTERP_2D
is a MATLAB library which
finds P(X,Y), a polynomial interpolant to data Z(X,Y) which depends on
two independent variables X and Y, by setting up and solving a linear
system involving the Vandermonde matrix.
This software is primarily intended as an illustration of the problems
that can occur when the interpolation problem is naively formulated
using the Vandermonde matrix. Unless the data points are well separated,
and the degree of the polynomial is low, the linear system will become
very difficult to store and solve accurately, because the monomials
used as basis vectors by the Vandermonde approach become indistinguishable.
If the data is available on a product grid, then both the LAGRANGE_INTERP_2D
and VANDERMONDE_INTERP_2D libraries will be trying to compute the same
interpolating function. However, especially for higher degree polynomials,
the Lagrange approach will be superior because it avoids the badly conditioned
Vandermonde matrix associated with the usage of monomials as the basis.
The Lagrange approach uses as a basis a set of Lagrange basis polynomials
l(i,j)(x) which are 1 at node (x(i),y(j)) and zero at the other nodes.
VANDERMONDE_INTERP_2D needs access to the QR_SOLVE and R8LIB libraries.
The test code also needs access to the TEST_INTERP_2D library.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
VANDERMONDE_INTERP_2D is available in
a C version and
a C++ version and
a FORTRAN77 version and
a FORTRAN90 version and
a MATLAB version.
Related Data and Programs:
LAGRANGE_INTERP_2D,
a MATLAB library which
defines and evaluates the Lagrange polynomial p(x,y)
which interpolates a set of data depending on a 2D argument
that was evaluated on a product grid,
so that p(x(i),y(j)) = z(i,j).
PWL_INTERP_2D,
a MATLAB library which
evaluates a piecewise linear interpolant to data defined on
a regular 2D grid.
R8LIB,
a MATLAB library which
contains many utility routines using double precision real (R8) arithmetic.
RBF_INTERP_2D,
a MATLAB library which
defines and evaluates radial basis function (RBF) interpolants to 2D data.
SHEPARD_INTERP_2D,
a MATLAB library which
defines and evaluates Shepard interpolants to 2D data,
which are based on inverse distance weighting.
TEST_INTERP_2D,
a MATLAB library which
defines test problems for interpolation of data z(x,y) of a 2D argument.
TOMS886,
a MATLAB library which
defines the Padua points for interpolation in a 2D region,
including the rectangle, triangle, and ellipse,
by Marco Caliari, Stefano de Marchi, Marco Vianello.
This is a MATLAB version of ACM TOMS algorithm 886.
VANDERMONDE_APPROX_2D,
a MATLAB library which
finds a polynomial approximant p(x,y) to data z(x,y) of a 2D argument by setting up and
solving an overdetermined linear system for the polynomial coefficients
involving the Vandermonde matrix.
VANDERMONDE_INTERP_1D,
a MATLAB library which
finds a polynomial interpolant to a function of 1D data
by setting up and solving a linear system for the polynomial coefficients,
involving the Vandermonde matrix.
Reference:

Kendall Atkinson,
An Introduction to Numerical Analysis,
Prentice Hall, 1989,
ISBN: 0471624896,
LC: QA297.A94.1989.

Philip Davis,
Interpolation and Approximation,
Dover, 1975,
ISBN: 0486624951,
LC: QA221.D33

David Kahaner, Cleve Moler, Steven Nash,
Numerical Methods and Software,
Prentice Hall, 1989,
ISBN: 0136272584,
LC: TA345.K34.
Source Code:
Examples and Tests:
The test code requires the test_interp_2d library as well. If this library is
available in a separate folder at the same "level" as the vandermonde_interp_2d library,
then a Matlab command such as "addpath ( '../test_interp_2d')" will make that library
accessible for a run of the test program.
The code generates some plots of the data and approximants.

p01_data.png,
a plot of the piecewise linear interpolant to the data for problem p01;

p01_poly01.png,
a plot of the polynomial interpolant for problem p01, degree 1;

p01_poly02.png,
a plot of the polynomial interpolant for problem p01, degree 2;

p01_poly04.png,
a plot of the polynomial interpolant for problem p01, degree 4;

p01_poly08.png,
a plot of the polynomial interpolant for problem p01, degree 8;

p02_data.png,
a plot of the piecewise linear interpolant to the data for problem p02;

p02_poly01.png,
a plot of the polynomial interpolant for problem p02, degree 1;

p02_poly02.png,
a plot of the polynomial interpolant for problem p02, degree 2;

p02_poly04.png,
a plot of the polynomial interpolant for problem p02, degree 4;

p02_poly08.png,
a plot of the polynomial interpolant for problem p02, degree 8;

p03_data.png,
a plot of the piecewise linear interpolant to the data for problem p03;

p03_poly01.png,
a plot of the polynomial interpolant for problem p03, degree 1;

p03_poly02.png,
a plot of the polynomial interpolant for problem p03, degree 2;

p03_poly04.png,
a plot of the polynomial interpolant for problem p03, degree 4;

p03_poly08.png,
a plot of the polynomial interpolant for problem p03, degree 8;

p04_data.png,
a plot of the piecewise linear interpolant to the data for problem p04;

p04_poly01.png,
a plot of the polynomial interpolant for problem p04, degree 1;

p04_poly02.png,
a plot of the polynomial interpolant for problem p04, degree 2;

p04_poly04.png,
a plot of the polynomial interpolant for problem p04, degree 4;

p04_poly08.png,
a plot of the polynomial interpolant for problem p04, degree 8;

p05_data.png,
a plot of the piecewise linear interpolant to the data for problem p05;

p05_poly01.png,
a plot of the polynomial interpolant for problem p05, degree 1;

p05_poly02.png,
a plot of the polynomial interpolant for problem p05, degree 2;

p05_poly04.png,
a plot of the polynomial interpolant for problem p05, degree 4;

p05_poly08.png,
a plot of the polynomial interpolant for problem p05, degree 8;

p06_data.png,
a plot of the piecewise linear interpolant to the data for problem p06;

p06_poly01.png,
a plot of the polynomial interpolant for problem p06, degree 1;

p06_poly02.png,
a plot of the polynomial interpolant for problem p06, degree 2;

p06_poly04.png,
a plot of the polynomial interpolant for problem p06, degree 4;

p06_poly08.png,
a plot of the polynomial interpolant for problem p06, degree 8;

p07_data.png,
a plot of the piecewise linear interpolant to the data for problem p07;

p07_poly01.png,
a plot of the polynomial interpolant for problem p07, degree 1;

p07_poly02.png,
a plot of the polynomial interpolant for problem p07, degree 2;

p07_poly04.png,
a plot of the polynomial interpolant for problem p07, degree 4;

p07_poly08.png,
a plot of the polynomial interpolant for problem p07, degree 8;

p08_data.png,
a plot of the piecewise linear interpolant to the data for problem p08;

p08_poly01.png,
a plot of the polynomial interpolant for problem p08, degree 1;

p08_poly02.png,
a plot of the polynomial interpolant for problem p08, degree 2;

p08_poly04.png,
a plot of the polynomial interpolant for problem p08, degree 4;

p08_poly08.png,
a plot of the polynomial interpolant for problem p08, degree 8;

p09_data.png,
a plot of the piecewise linear interpolant to the data for problem p09;

p09_poly01.png,
a plot of the polynomial interpolant for problem p09, degree 1;

p09_poly02.png,
a plot of the polynomial interpolant for problem p09, degree 2;

p09_poly04.png,
a plot of the polynomial interpolant for problem p09, degree 4;

p09_poly08.png,
a plot of the polynomial interpolant for problem p09, degree 8;

p10_data.png,
a plot of the piecewise linear interpolant to the data for problem p10;

p10_poly01.png,
a plot of the polynomial interpolant for problem p10, degree 1;

p10_poly02.png,
a plot of the polynomial interpolant for problem p10, degree 2;

p10_poly04.png,
a plot of the polynomial interpolant for problem p10, degree 4;

p10_poly08.png,
a plot of the polynomial interpolant for problem p10, degree 8;

p11_data.png,
a plot of the piecewise linear interpolant to the data for problem p11;

p11_poly01.png,
a plot of the polynomial interpolant for problem p11, degree 1;

p11_poly02.png,
a plot of the polynomial interpolant for problem p11, degree 2;

p11_poly04.png,
a plot of the polynomial interpolant for problem p11, degree 4;

p11_poly08.png,
a plot of the polynomial interpolant for problem p11, degree 8;

p12_data.png,
a plot of the piecewise linear interpolant to the data for problem p12;

p12_poly01.png,
a plot of the polynomial interpolant for problem p12, degree 1;

p12_poly02.png,
a plot of the polynomial interpolant for problem p12, degree 2;

p12_poly04.png,
a plot of the polynomial interpolant for problem p12, degree 4;

p12_poly08.png,
a plot of the polynomial interpolant for problem p12, degree 8;

p13_data.png,
a plot of the piecewise linear interpolant to the data for problem p13;

p13_poly01.png,
a plot of the polynomial interpolant for problem p13, degree 1;

p13_poly02.png,
a plot of the polynomial interpolant for problem p13, degree 2;

p13_poly04.png,
a plot of the polynomial interpolant for problem p13, degree 4;

p13_poly08.png,
a plot of the polynomial interpolant for problem p13, degree 8;
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the MATLAB source codes.
Last modified on 02 August 2012.